Can you become the high king or Emperor in Skyrim? I have seen a few people discuss this and most say you don't need to because you are Dragonborn, but most of the individuals in the past who have been dragonborn including Remen and Talos took the role of leadership within the Empire.

In the current game, no.

In DLC to come? I guess we'll have to wait and find out.

I doubt it. The player character in these games is never anyone of lasting importance, or every game would end with a Dragon Break

I feel like the dragon breaks are a cop out by Beth, and not allowing your PC to make lasting changes in the world makes the game a lot less appealing. Sure you killed Alduin, but what happens to your character afterwards. The Nerevarine heads to akavir and is never heard from again....lame

Personally a DLC/XPAC in which we become the High King or Emperor AND get to fight the Thalmor seems extremely appealing to me (and others).

Or you could join the Thalmor side, become a high ranking Thalmor official (or their leader if you're an Altmer) and destroy the Empire/Skryim.

The possibilities are high.